Abstract:

Imagine mathematics as a large island continent, rather like Australia, surrounded by the seas of the sciences it serves: Physics, Chemistry, Economics, Engineering and so on. The continent is divided into states: Algebra, Complex Analysis, Topology and the rest, which are further divided into counties representing more specialized topics. Fixed point theory is not a territory in this continent, rather it is a river that flows through parts of it. The headwaters of the river lie in Topology, coming from both algebraic and geometric regions, it then meanders through many counties of Functional Analysis, of both the classical and nonlinear type, before moving on to Differential Equations from which it flows through Applied Mathematics on its way to several of the seas. We will travel along this river starting with an application and sailing upstream, docking briefly at several states along the way. Of course our boat can only skim the surface of the fixed point theory river, but our trip back to its source may throw a little light on the vexing question "why should anyone other than a topology specialist study algebraic topology?".